Three-dimensional Brownian loop-soup cluster dimension conjecture

Let $d=3$, and let the cable-graph loop-soup clusters be formed on finer and finer meshes. Denote by $\Delta$ the Hausdorff dimension of the Brownian loop-soup clusters. Three-dimensional cluster conjecture. The scaling limit of the cable-graph loop-soup clusters in three dimensions should be exactly the collection of Brownian loop-soup clusters, and

$$\Delta=\frac{5}{2}.$$

In two dimensions the corresponding scaling limit is known to equal the Brownian loop-soup clusters. In three dimensions, the cluster dimension is known to be greater than $2$ and at most $5/2$, but the asserted scaling limit and exact dimension remain open.